Order of accuracy

In numerical analysis, order of accuracy quantifies the rate of convergence of a numerical approximation of a differential equation to the exact solution. Consider ${\displaystyle u}$, the exact solution to a differential equation in an appropriate normed space ${\displaystyle (V,||\ ||)}$. Consider a numerical approximation ${\displaystyle u_{h}}$, where ${\displaystyle h}$ is a parameter characterizing the approximation, such as the step size in a finite difference scheme or the diameter of the cells in a finite element method. The numerical solution ${\displaystyle u_{h}}$ is said to be ${\displaystyle n}$th-order accurate if the error ${\displaystyle E(h):=||u-u_{h}||}$ is proportional to the step-size ${\displaystyle h}$ to the ${\displaystyle n}$th power:^[1]

${\displaystyle E(h)=||u-u_{h}||\leq Ch^{n}}$

where the constant ${\displaystyle C}$ is independent of ${\displaystyle h}$ and usually depends on the solution ${\displaystyle u}$.^[2] Using the big O notation an ${\displaystyle n}$th-order accurate numerical method is notated as

${\displaystyle ||u-u_{h}||=O(h^{n})}$

This definition is strictly dependent on the norm used in the space; the choice of such norm is fundamental to estimate the rate of convergence and, in general, all numerical errors correctly.

The size of the error of a first-order accurate approximation is directly proportional to ${\displaystyle h}$. Partial differential equations which vary over both time and space are said to be accurate to order ${\displaystyle n}$ in time and to order ${\displaystyle m}$ in space.^[3]